1. 5.1 Basic Probability Ideas
Definition: Experiment – obtaining a piece of data
Definition: Outcome – result of an experiment
Definition: Sample space – list of all possible outcomes of an experiment
Definition: Event – collection of outcomes from an experiment (a simple event is a single outcome)
Definition: Equally likely sample space – sample space in which all outcomes are equally likely

2. 5.1 Basic Probability Ideas
Definition: Probability – A number between 0 and 1 (inclusive) that indicates how likely an event is to occur
Definition: n(A) – number of simple events in A
Definition: Theoretical probability of A – p(A) = n(A)/n(S) = (# outcomes of A) ÷ (# outcomes in the equally likely sample space)

3. 5.1 Basic Probability Ideas
Law of Large Numbers – as the number of experiments increases without bound, the proportion of a certain event approaches a theoretical probability
The Law of Large Numbers does not say: If you throw a coin and get heads 10 times, that your probability of getting tails increases. P(heads) stays at 1/2

4. 5.1 Basic Probability Ideas
Empirical Probability – relative frequency of an event based on past experience p(A) = (# times A has occurred) ÷ (# observations)

5. 5.1 Basic Probability Ideas
Properties of Probabilities:
0 ≤ p(A) ≤ 1
p(a) = 0  A is impossible, the event will never happen
p(a) = 1  A is a certain event, the event must happen
Let a1, a2, a3,…. an be all events in a sample space, then p(a1) + p(a2) + … + p(an) = 1

6. 5.2 Rules of Probability
Definition: Complement of an event A – The event that A does not occur denoted AC
Properties:
P(A) + p(AC) = 1
P(AC) = 1 – p(A)
P(A) = 1 – p(AC)
Odds in favor of event A – n(A):n(AC) or n(A) ÷ n(AC)

7. 5.2 Rules of Probability
Example: p(A) = 1/3 then p(AC) = 1 – p(A) = 2/3 odds in favor of A = p(A):p(AC) = 1/3:2/3 = 1:2 odds against A = p(AC):p(A) = 2/3:1/3 = 2:1
Probabilities from odds in favor – odds in favor = s:f (successes to failures) p(A) = s/(s + f) p(AC) = f/(s + f)

8. 5.2 Rules of Probability
Joint probability tables – displays possible outcomes and their likelyhood of occurrence
Example: Given the following table of data:
Coke
Pepsi
Male 13 31
Female 22 14

9. 5.2 Rules of Probability
Probability table for example (total of 80 people in the sample):
Coke
Pepsi
Male
13/80 = .1625
31/80 = .3875
Female
22/80 = .275
14/80 = .175

10. 5.2 Rules of Probability B G Simple probability tree: Boy branches
root G
Girl

11. 5.2 Rules of Probability
Probability trees are useful when events do not have the same probability (there is no equally likely sample space)
Problem solutions involving trees can become long if many branches are to be calculated (similar to the brute force method in section 4.5)

12. 5.3 Probabilities of Unions and Intersections
Definition: The union of two events A and B is the event that occurs if either A or B or both occur in a single experiment. The union of A and B is denoted A  B Example: (rolling a die – getting an even number or a perfect square) 2 4 6 5 3 1

13. 5.3 Probabilities of Unions and Intersections
Definition: The intersection of two events A and B is the event that occurs if both A and B occur in a single experiment. The intersection of A and B is denoted A  B Example: (rolling a die – getting an even number and a perfect square) 2 4 6 5 3 1

14. 5.3 Probabilities of Unions and Intersections
Definition: mutually exclusive or disjoint events – events for which A  B =  (where  represents an event with no elements)
If A and B are mutually exclusive, then:
P(A  B) = 0
P(A  B) = P(A) + P(B)
Union Principle of Probability: P(A  B) = P(A) + P(B) - P(A  B)

15. 5.4 Conditional Probability and Independence
Definition: The conditional probability of A given B is the probability of A occurring given that B has already occurred – denoted P(AB) When outcomes are equally likely:
n(AB)
n(B)
P(AB) =
Conditional Probability Formula (outcomes not necessarily equally likely)
P(AB)P(B)
P(AB) =

16. 5.4 Conditional Probability and Independence
Multiplication Principal: P(A  B) = P(B)  P(AB)
Tree diagrams – useful for conditional probability because each section of a branch is a probability conditional by the previous branches

17. 5.4 Conditional Probability and Independence
Independence: Two events A and B are said to be independent if the occurrence of A does not affect P(B) and vice versa. A & B are independent if: P(AB) = P(A) or P(BA) = P(B)
Multiplication Principle for Independent Events: A & B are independent events  P(A  B) = P(A)  P(B)

18. P(A)  P(BA) + P(AC)  P(BAC)
5.5 Bayes’ Formula
Bayes formula for 2 cases:
P(A)  P(BA)
P(AB) =
P(A)  P(BA) + P(AC)  P(BAC)

19. P(A1)  P(BA1) + P(A2)  P(BA2) + … + P(An)  P(BAn)
5.5 Bayes’ Formula
Bayes formula for n disjoint events:
P(Ai)  P(BAi)
P(AiB) =
P(A1)  P(BA1) + P(A2)  P(BA2) + … + P(An)  P(BAn)

20. 5.6 Permutations and Combinations
Multiplication Principle – given a tree with the number of choices at each branch being m1, m2, m3, … mn, then the number of possible occurrences is: m1  m2  m3  …  mn

21. 5.6 Permutations and Combinations
Permutations: The number of arrangements of r items from a set of n items. Note: Order matters. n!
(n – r)!
nPr =

22. 5.6 Permutations and Combinations
Combinations: The number of subsets of r items from a set of n items. Note: Order does not matter. n!
(n – r)!  r!
nCr =

23. 5.6 Permutations and Combinations - summary of counting formulas
Without replacement
With replacement
(order matters)
Order does not matter
(subsets)
Order matters
(arrangements)
Multiplication principal
Permutation
Combination

24. 5.7 Probability and Counting Formulas
Example: A bag contains 4 red marbles and 3 blue marbles. Find the probability of selecting:
Two red marbles
Two blue marbles
A red marble followed by a blue marble
# ways to pick 2 red # ways to pick any 2 marbles
= 4C2
7C2
P(2 red) =
= 6/21 = 2/7

25. 5.7 Probability and Counting Formulas
# ways to pick 2 blue # ways to pick any 2 marbles
P(2 blue) =
= 3C2
7C2
= 3/21 = 1/7
chance of picking red on first  chance of picking blue on second
P(red then blue) =
= 4/7  3/6 = 2/7

26. 5.7 Probability and Counting Formulas
Birthday Problem: Suppose there are n people in a room. Find the formula for the probability that at least two people have the same birthday. Note: P(at least two birthdays the same) = 1 – P(no two birthdays the same) # ways for n people to have birthdays = 365n # ways for for n birthdays without repeats = 365Pn answer = 1 – (365Pn  365n)

27. 5.8 Expected Value
Expected Value – for a given sample space with disjoint outcomes having probabilities p1, p2, p3, … pn and a value (winnings) of x1, x2, x3, … xn , then the expected value of the sample space is: x1 p1 + x2 p2 + x3 p3 +…. xn pn
Definition: A game is said to be fair if the cost of participating equals the expected winnings.
Expected winnings < cost  unfair to participant
Expected winnings > cost  unfair to organizers

28. 5.9 Binomial Experiments Definition: Binomial Experiment
The same trial is repeated n times.
There are only 2 possible outcomes for each trial – success or failure
The trials are independent so the probability of success or failure is the same for each trial.

29. 5.9 Binomial Experiments
Binomial Probabilities: P(x successes) = nCr  px  (1-p)n-x with x = 0,1,2,…,n where n is the number of trials, p is the probability of success, and x is the number of successful trials
Expected value of a binomial experiment = n  p note: The most likely outcome “tends to be close to” the expected value.